Icosahedral polyhedra from d6 lattice and danzer’s abck tiling

Abeer Al-Siyabi, Nazife Ozdes Koca*, Mehmet Koca

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

It is well known that the point group of the root lattice D6 admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group H3, its roots, and weights are determined in terms of those of D6. Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by projections of the sets of lattice vectors of D6 determined by a pair of integers (m1, m2) in most cases, either both even or both odd. Vertices of the Danzer’s ABCK tetrahedra are determined as the fundamental weights of H3, and it is shown that the inflation of the tiles can be obtained as projections of the lattice vectors characterized by the pair of integers, which are linear combinations of the integers (m1, m2) with coefficients from the Fibonacci sequence. Tiling procedure both for the ABCK tetrahedral and the <ABCK> octahedral tilings in 3D space with icosahedral symmetry H3, and those related transformations in 6D space with D6 symmetry are specified by determining the rotations and translations in 3D and the corresponding group elements in D6. The tetrahedron K constitutes the fundamental region of the icosahedral group and generates the rhombic triacontahedron upon the group action. Properties of “K-polyhedron”, “B-polyhedron”, and “C-polyhedron” generated by the icosahedral group have been discussed.

Original languageEnglish
Article number1983
Pages (from-to)1-15
Number of pages15
JournalSymmetry
Volume12
Issue number12
DOIs
Publication statusPublished - Dec 2020
Externally publishedYes

Keywords

  • Aperiodic tilings
  • Coxeter–Weyl groups
  • Icosahedral group
  • Lattices
  • Polyhedra
  • Projections of polytopes
  • Quasicrystals

ASJC Scopus subject areas

  • Computer Science (miscellaneous)
  • Chemistry (miscellaneous)
  • Mathematics(all)
  • Physics and Astronomy (miscellaneous)

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